determination of the equivalent anisotropy properties of ... · g. mingione (2006), a. de simone,...

Determination of the equivalent anisotropyproperties of polycrystalline magnetic materials:

theoretical aspects and numerical analysis

Michela Eleuteri

University of Verona (Italy)

Supported by: GNAMPA 2010 “Problemi variazionali inmicromagnetismo” (resp. M. Eleuteri)

Joint project with...

. Variational problems in micromagnetism

I. Fonseca, G. Leoni, (2000); G. Pisante (2004); E. Acerbi, I. Fonseca,G. Mingione (2006), A. De Simone, R.V. Kohn, S. Müller, and F. Otto(2006).

. Equivalent anisotropy properties of polycrystalline magneticmaterials

Oriano Bottauscio and Alessandra Manzin (INRIM - Istituto Nazionale diRicerca Metrologica)

Valeria Chiadó Piat (Politecnico di Torino)

Luca Lussardi (Università Cattolica di Brescia)

. Hybrid magnetic nanostructures for field detection

Marco Coïsson and Alessandra Manzin (INRIM - Istituto Nazionale diRicerca Metrologica)

Sambri Alessia (Università degli Studi di Napoli “Federico II”).

Michela Eleuteri Polycrystalline magnetic materials

Task of the research project

Development of theoretical formulations based on the classical tools ofMathematical Analysis, in particular the methods of the Calculus ofVariations and of the analysis of Partial Differential Equations, forstudying magnetization processes in complex magnetic nanostructures

Strong connection between PoliTO and INRIM, between mathematicsand numerics, with the aim of giving the necessary mathematicalsupport for developing advanced computational models for themicromagnetic calculus and the analysis of the nanostructuresproduced the INRIM research team

The relevance of such activity is underlined by the more and moregrowing attention which has been dedicated, during the last decade, tothe study of micromagnetic problems with the methods of the Calculusof Variations and, more generally, of the Mathematical Analysis

The basic aim is the comprehension of the connections between thematerial properties on the microscopic level and the behavior on themacroscopic scale, with the idea of giving a modeling interpretation ofthe experimental data

The attention is focused on the study of the Landau-Lifschitz-Gilbert(LLG) equation, universally adopted in the prediction of the nonlineardynamic of magnetic materials on micron- and submicron-scaleGreat importance to the analysis of the minima of the Gibbs free energyfunctional, strictly connected to the LLG equation, with the aim ofdetermining the equilibrium configurations in complex magnetic systemsThe development of mathematical models for the analysis of magneticnanostructures may present criticalities concerned with the treatment ofthe spatial distribution of the physical and structural parameters, whosecombination gives origin to phenomena characterized by a stronglymultiscale natureIn particular, the energy functional is characterized by the simultaneouspresence of energetic terms that deal with very different length scales,whose interaction must be treated in a very accurate way, in order tohave a correct interpretation of the evolution of magnetic domainsTo this end, a first aim will be concerned with the development ofmultiscale formulations based on homogenization techniques and on theGamma-convergence theory

Plan of the talk

Nanostructures and nanotechnology

Magnetic nanostructures (examples and polycrystalline magneticmaterials)

Criticalities from the strongly multiscale nature of the magnetizationprocesses (numerical treatment difficult task)

Our approach and our results

Plan of the talk

Nanostructures and nanotecnology

General agreement of the fact that the first person who had intuitionabout the potentialities of nanotecnologies was R. Feyman (Nobel Prizein Physics, in 1965)

In a well known conference in 1959 at the Californian Institute ofTechnology said a sentence that became famous: “There is plenty ofroom at the bottom” (meaning “at the atomic dimension”)

Feyman intuition was that it could have been possible to manipulate thematter at the atomic or subatomic scale

Figure 1: A nickel sheet where there have been posed atoms singularlymanipulated (From: Marco Bettinelli “Nanostrutture enano-tecnologie”(2008))

Nanotechnology: to modify intentionally matter, at the atomic orsubatomic scale (interdisciplinary)

Figure 1: A nickel sheet where there have been posed atoms singularlymanipulated (From: Marco Bettinelli “Nanostrutture enano-tecnologie”(2008))

Nanotechnology: to modify intentionally matter, at the atomic orsubatomic scale (interdisciplinary)

Magnetic nanostructures

R. Skomski “Nanomagnetis” J. Phys.: Condens. Matter, 15 (2003)R841-R896.

Thousands of years of human curiosity have led to the discovery ofmagnetism, and for many centuries magnetism has stimulated progressin science and technology

For a long time, the focus had been on macroscopic magnetism, asexemplified by the compass needle, by the geomagnetic field and by theability of electromagnets and permanent magnets to do mechanicalwork

Atomic-scale magnetic phenomena, such as quantum-mechanicalexchange, crystal-field interaction or relativistic spin-orbit coupling werediscovered in the first half of the last century and are now exploited, forexample, in advanced permanent- magnet intermetallics

However, only in recent decades it became clear that solid-statemagnetism is, to a large extent, a nanostructural phenomenon

The scientific and technological importance of magnetic nanostructureshas different reasons (overwhelming variety of structures with interestingphysical propertiesc - from naturally occurring nanomagnets andcomparatively easy-to-produce bulk nanocomposites to demandingartificial nanostructures; involvement of nanoscale effects in theexplanation and improvement of the properties of advanced magneticmaterials; the fact that nanomagnetism opened the door for completelynew technologies)

A naturally occurring biomagnetic phenomenon is magnetite (Fe3O4)nanoparticles precipitated in bacteria, molluscs, insects and higheranimals

Magnetostatic bacteria live in dark environments and contain chains of40-100 nm magnetite particles used for vertical orientation

Similar magnetite particles have been found in the brains of bees,pigeons and tuna, and it is being investigated whether and how theparticles serve as field sensors for migration

Magnetite and other oxide particles are also responsible for rockmagnetism, exploited for example in archaeomagnetic dating and formonitoring changes in the Earth’s magnetic field

A fascinating approach is artificial nanostructuring to create completelynew materials and technologies

Advanced magnetic nanostructures are characterized by a fascinatingdiversity of geometries, ranging from complex bulk structures to a broadvariety of low-dimensional systems - Figure 2 shows some examples

Figure 2: Typical nanostructure geometries: (a) chain of fine particles,(b) striped nanowire, (c) cylindrical nanowire, (d) nanojunction, (e) vicinalsurface step, (f) nanodots, (g) antidots and (h) particulate medium.

Micromagnetic numerical models: difficulties

Micromagnetic numerical models common tool for evaluating themagnetization processes in magnetic nanostructuresDeep understanding of the relationship between structural propertiesand magnetic domain formation - outcomes in technological applicationsMicromagnetics theory based on a continuum approximation of themagnetization spatial distribution - different energy contributionscompete to determine magnetic domain configurationMost important contributions: exchange, magnetostatic, anisotropy andZeeman energies whose accurate approximation is essential to correctlyevaluate time and spatial evolution of magnetization within the sampleEnergy terms act at different spatial scales (nanometric resolution:exchange term; sample dim.: long-range magnetostatic interactions)Magnetic anisotropy expresses the tendency of the magnetization to liealong certain crystallographic directions - intermediate spatial scaleMagnetic properties are influenced by the size, shape, boundaryproperties and orientation distribution of grainsMicromagnetic simulations complex task - simulations often performedunder the hypothesis of single crystal

Assumption of uniaxial or cubic anisotropy (valid for samples havingdimensions comparable than grain size)

Assumption inadequate in presence of polycrystalline materials (grainorientation and distribution affect the behaviour of the system)

Polycrystalline materials usually modelled within micromagneticnumerical codes as an array of grains, geometrically constructed usingVoronoi diagrams - each single-crystal grain is assumed to have arandomly oriented uniaxial anisotropy

Strong impact on the geometry building and meshing, making thepre-processing phase very complex

We propose an alternative procedure, based on the replacement of thepolycrystalline magnetic material with an homogenized sample havingequivalent anisotropy properties

Magnetic medium modeled as an assembly of monocrystalline grains,assuming a stochastic spatial distribution of easy axes

The mathematical theory of Γ-convergence is then applied tohomogenize the anisotropic term in the Gibbs free energy

First results obtained

O. BOTTAUSCIO, V. CHIADÒ PIAT, M. ELEUTERI, L. LUSSARDI, A. MANZIN:“Determination of the equivalent anisotropy properties of polycrystallinemagnetic materials: theoretical aspects and numerical analysis”, submitted.

The Γ−convergence theory is here adopted to homogenize theanisotropic contribution in the energy functional and derive theequivalent anisotropy propertiesThe reliability of this approach is investigating focusing on thecomputation of the static hysteresis loops of polycrystalline magneticthin films, starting from the numerical integration of the LLG equation

O. BOTTAUSCIO, V. CHIADÒ PIAT, M. ELEUTERI, L. LUSSARDI, A. MANZIN:“Homogenization of random anisotropy properties in polycrystalline magneticmaterials”, “Proceedings of the 8th HMM”, Phys. B: Cond. Matt., (2012), DOI:10.1016/j.physb.2011.06.085.

Focus on the micromagnetic computations of reversal processes inpolycrystalline magnetic thin films (study of precessional switching,comparing the results for heterogeneous and homogenized media)

Outline

Homogenization results

Determination of equivalent anisotropy properties

Numerical validation of the homogenization results: computation ofswitching processes and of the static hysteresis loops in polycrystallinemagnetic thin films

Outline

Setting of the problem

Polycrystalline magnetic sample, occupying a bounded open regionD⊂ R3

M ∈ L2(R3;R3) magnetization vector field, with M = MSχD in R3, MSbeing the saturation magnetization; m ∈ L2(R3;R3) m = M/MSrescaled magnetization

magnetic energy behaviour described by the following functional

F(m,Ha) =∫

DA|∇m|2 dx+

fan(m,uan)dx

−µ0

MSHm ·mdx−µ0

MSHa ·mdx

Hm = ∇u where ∇2u+∇ ·M = 0 in R3

More general setting for the anisotropy energy function: stochasticdistribution of easy axes

F(m,Ha) =∫

DA|∇m|2 dx+

fan(m,uan)dx

−µ0

MSHm ·mdx−µ0

MSHa ·mdx

F(m,Ha) =∫

DA|∇m|2 dx+

fan(m,uan)dx

−µ0

MSHm ·mdx−µ0

MSHa ·mdx

F(m,Ha) =∫

DA|∇m|2 dx+

fan(m,uan)dx

−µ0

MSHm ·mdx−µ0

MSHa ·mdx

F(m,Ha) =∫

DA|∇m|2 dx+

fan(m,uan)dx

−µ0

MSHm ·mdx−µ0

MSHa ·mdx

F(m,Ha) =∫

DA|∇m|2 dx+

fan(m,uan)dx

−µ0

MSHm ·mdx−µ0

MSHa ·mdx

fan(m,uan) = kan[1− (m ·uan)2]

F(m,Ha) =∫

DA|∇m|2 dx+

fan(m,uan)dx

−µ0

MSHm ·mdx−µ0

MSHa ·mdx

fan(m,uan) = kan[1− (m ·uan)2]

F(m,Ha) =∫

DA|∇m|2 dx+

fan(m,uan)dx

−µ0

MSHm ·mdx−µ0

MSHa ·mdx

fan(m,uan) = kan[1− (m ·uan(Txω))2]

Fε(m,ω) =∫

DA|∇m|2 dx+

xε,ω)

−µ0

MSHm ·mdx−µ0

MSHa ·mdx

fan(m,uan) = kan[1− (m ·uan(Txω))2]

The Γ−convergence result

m ∈ L2(R3;R3) : m|D ∈ H1(D;R3), |m|= χD a.e. in R3

For any m, let v ∈ H1(R3) be the unique solution of the equation

∇2v+∇ ·m = 0, in D ′(R3)

(Ω,F ,µ) probability space; T 3-dimensional ergodic dynamical system

Let ϕ : R3×R3×Ω→ [0,+∞) be a measurable map such thatϕ(0,0, ·) ∈ L1(Ω) and